Hermitian matrices

[diogomgf]

\(\displaystyle A_{n} \in \mathbb{C} \) is a square hermitian matrix.
\(\displaystyle \alpha \in \mathbb{C} \) is a scalar.

Show a necessary and sufficient condition for the matrix \(\displaystyle \alpha.A \) to be hermitian.

My thought process so far:

\(\displaystyle \alpha = a+ bi \) .
\(\displaystyle (A)_{i,j} = (A^*)_{i,j} = c + di \) .

\(\displaystyle ((\alpha.A)^*)_{i,j} = (\alpha.A)_{i,j} \) .
\(\displaystyle (\alpha.A)_{i,j} = ac + bci + adi - bc \) .
\(\displaystyle ((\alpha.A)^*)_{i,j} = ac - bci - adi - bc \) .

Don't know where to go from here...

 

[Romsek]

You're making it harder than it needs to be.

\(\displaystyle \text{$A$ is hermitian there for $A_{i,j}=A_{j,i}^*$}\)
\(\displaystyle (\alpha A_{j,i} )^* = \alpha^* A_{j,i}^*\)
\(\displaystyle (\alpha A)_{i,j} = (\alpha A)_{j,i}^* \Rightarrow \alpha A_{i,j} = \alpha^* A_{j,i}^*\)
\(\displaystyle \text{Thus $\alpha = \alpha^* \Rightarrow \alpha \in \mathbb{R}$}\)

I leave the other direction to you.

 

[diogomgf]

You're making it harder than it needs to be.

\(\displaystyle \text{$A$ is hermitian there for $A_{i,j}=A_{j,i}^*$}\)
\(\displaystyle (\alpha A_{j,i} )^* = \alpha^* A_{j,i}^*\)
\(\displaystyle (\alpha A)_{i,j} = (\alpha A)_{j,i}^* \Rightarrow \alpha A_{i,j} = \alpha^* A_{j,i}^*\)
\(\displaystyle \text{Thus $\alpha = \alpha^* \Rightarrow \alpha \in \mathbb{R}$}\)

I leave the other direction to you.

Thank you for the clear explanation